The value of sin (pi/3) equals the value of?

Question

A) sin (2pi/3)
B) sin (-pi/3)
C) sin (-5/6)
D) sin (5pi/6)
E) sin (-2pi/3)

Answer 1

A) sin 2pi/3 its the same angle in the 1st and 2nd quadrants

Answer 2

The answer is A.

There are several ways to show this, so here are a couple:

First of all, we know the sine refers to the height in the unit circle, and these two angles are at the same height of √3/2.

However, a much more impressive way to PROVE this is with a sum angle formula. The following is ALWAYS true:

sin(x+y) = sin(x)cos(y) + cos(x)sin(y)

So we know that 2pi/3 = pi/3 + pi/3, so make these x and y.

Therefore,

sin(pi/3 + pi/3) = sin(pi/3)*cos(pi/3) + cos(pi/3)+sin(pi/3),

and we know that cos(pi/3) is 1/2, so we have

sin(2pi/3) = (1/2)sin(pi/3) + (1/2)sin(pi/3) = sin(pi/3)

Answer 3

pi/3 is a third of the way to 180 degrees or pi. 1/3 of 180 is 60 degrees. Sin(60) is sqrt(3)/2. That's only possible in the first and forth quadrint. That means 60 degrees below 0 or in th negative direction is equal to pi/3.

B. sin(-pi/3)

Answer 4

A) sin (2pi/3)

Answer 5

A

NOT B
sin(-pi/3) = -sin(pi/3).

Answer 6

A